# Scalene Triangle

In geometry, Scalene Triangle is a triangle that has all its sides of different lengths.  It means all the sides of a scalene triangle are unequal and all the three angles are also of different measures. This is one of the three types of triangles, based on sides.

We are going to discuss here its definition, formulas for perimeter and area and its properties. The triangles are defined based on their sides and angles. In geometry, a triangle is a closed two-dimensional plane figure with three sides and three angles and is shown as a three-sided polygon. It has three vertices and three edges.

## Definition

A scalene triangle is a triangle in which all three sides are in different lengths, and all three angles are of different measures. However, the sum of all the interior angles is always equal to 180 degrees. Thus, it meets the angle sum property condition of triangle.

### Classification of Triangles

Based on the sides and the interior angles of a triangle, there are different types of triangle. According to the interior angles of the triangle, it can be classified into three types, namely:

According to the sides of the triangle, the triangle can be classified into three types, namely;

## Properties

Some of the important properties of the scalene triangle are as follows:

• It has no equal sides.
• It has no equal angles.
• It has no line of symmetry.
• It has no point symmetry.
• The angles inside this triangle can be an acute, obtuse or right angle.
• If all the angles of the triangle are less than 90 degrees(acute), then the center of the circumscribing circle will lie inside a triangle.
• In a scalene obtuse triangle, the circumcenter will lie outside the triangle.
• A scalene triangle can be an obtuse-angled, acute-angled or right-angled triangle.

### Comparison-Scalene, Isosceles and Equilateral

 Scalene Triangle Isosceles Triangle Equilateral Triangle All three sides are unequal Any two sides are equal All three sides are equal All three angles are different Angles opposite to equal sides are equal All three angles equal to 60 degrees

## Formulas (Area and Perimeter)

Let us learn the formulas to find the area and perimeter of a triangle that has unequal sides and angles, or we can say the scalene triangle.

### Area of a Scalene Triangle

Area  = (1/2) x b x h square units

Where,

“b” refers to the base of the triangle

“h” refers to the height of the triangle

If the sides of the triangle are given, then apply Heron’s formula.

Area of the triangle =  $$A = \sqrt{s (s-a)(s-b)(s-c)}$$ square units

Where, s is the semi perimeter of a triangle, which can be found using the formula

s = (a+b+c)/2

Here,

a, b, and c denotes the sides of the triangle

### Perimeter of a Scalene Triangle

The perimeter of a triangle is equal to the sum of the length of sides of a triangle and it is given as:

Perimeter  = a + b + c units

Example: Consider a given triangle

To find the perimeter for the given triangle, add the sides of a triangle

Therefore, perimeter = 15 + 34 + 32 = 81 cm

## Solved Problems

Ques.1: Find the area of the scalene triangle ABC with the sides 8 cm, 6 cm and 4 cm

Solution:

Let a= 8 cm

b = 6 cm

c = 4 cm

If all the sides of a triangle are given, then use Heron’s formula.

Area of triangle =  $$\sqrt{s (s-a)(s-b)(s-c)}$$

Now, find the semiperimeter value

s = (a+b+c)/2

s = (8+6+4)/2

s = 18/2

s = 9

Now substitute the value of S in the area formula,

Area = $$\sqrt{9(9-8)(9-6)(9-4)}$$

$$=\sqrt{9(1)(3)(5)}$$

=√135

=11.6

Therefore, the area of the scalene triangle = 11.6 cm2

Ques.2: If the sides of a triangle are 9cm, 13 cm and 14cm. Find its perimeter.

Solution: Given, length of sides of the triangle are 9cm, 13cm and 14cm.

Perimeter = 9 + 13 + 14 = 36 cms

### Practice Questions

Find the area and perimeter of a triangle, for the length of three sides, given below:

• 3cm, 4cm and 5cm
• 8 in., 4in. and 12in
• 3cm, 6cm and 8cm
• 8cm, 11cm and 13cm

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## Frequently Asked Questions – FAQs

### What is a scalene triangle?

A scalene triangle is a triangle that has all its sides unequal in length and all its angles unequal in measure.

### What is the unique property of the scalene triangle?

Scalene triangle has all sides unequal. It has no line of symmetry. Interior angles can be acute, obtuse or right-angle.

### Does the scalene triangle obey the angle-sum property?

Scalene triangle has all its three angles different in measures and obeys the angle sum property, i.e. the sum of the three interior angles equals 180 degrees.

### What is a right-scalene triangle?

When one of the three angles measure 90 degrees and the angles or lengths of other two sides are not congruent, then the scalene triangle is called right scalene triangle.

### What is the area and perimeter of scalene triangle?

Area of scalene triangle is equal to half of the product of its base-length and height.
Perimeter is equal to the sum of its three unequal sides.

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